Revista Matemática Iberoamericana

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Volume 33, Issue 1, 2017, pp. 29–66
DOI: 10.4171/RMI/927

Published online: 2017-02-22

Computing minimal interpolants in $C^{1,1}(\mathbb R^d)$

Ariel Herbert-Voss[1], Matthew J. Hirn[2] and Frederick McCollum[3]

(1) Harvard University, Cambridge, USA
(2) Michigan State University, East Lansing, USA
(3) New York University, USA

We consider the following interpolation problem. Suppose one is given a finite set $E \subset \mathbb R^d$, a function $f \colon E \to \mathbb R$, and possibly the gradients of $f$ at the points of $E$. We want to interpolate the given information with a function $F \in C^{1,1}(\mathbb R^d)$ with the minimum possible value of Lip$(\nabla F)$. We present practical, efficient algorithms for constructing an $F$ such that Lip$(\nabla F)$ is minimal, or for less computational effort, within a small dimensionless constant of being minimal.

Keywords: Algorithm, interpolation, Whitney extension, minimal Lipschitz extension

Herbert-Voss Ariel, Hirn Matthew, McCollum Frederick: Computing minimal interpolants in $C^{1,1}(\mathbb R^d)$. Rev. Mat. Iberoamericana 33 (2017), 29-66. doi: 10.4171/RMI/927